43 research outputs found
Algorithms for outerplanar graph roots and graph roots of pathwidth at most 2
Deciding whether a given graph has a square root is a classical problem that
has been studied extensively both from graph theoretic and from algorithmic
perspectives. The problem is NP-complete in general, and consequently
substantial effort has been dedicated to deciding whether a given graph has a
square root that belongs to a particular graph class. There are both
polynomial-time solvable and NP-complete cases, depending on the graph class.
We contribute with new results in this direction. Given an arbitrary input
graph G, we give polynomial-time algorithms to decide whether G has an
outerplanar square root, and whether G has a square root that is of pathwidth
at most 2
Induced Disjoint Paths in Circular-Arc Graphs in Linear Time
The Induced Disjoint Paths problem is to test whether a graph G with k distinct pairs of vertices (si,ti) contains paths P1,…,Pk such that Pi connects si and ti for i=1,…,k, and Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their ends) for 1≤
Graph editing to a given degree sequence
We investigate the parameterized complexity of the graph editing problem called Editing to a Graph with a Given Degree Sequence where the aim is to obtain a graph with a given degree sequence σ by at most k vertex deletions, edge deletions and edge additions. We show that the problem is W[1]-hard when parameterized by k for any combination of the allowed editing operations. From the positive side, we show that the problem can be solved in time 2O(k(Δ⁎+k)2)n2logn for n -vertex graphs, where Δ⁎=maxσ, i.e., the problem is FPT when parameterized by k+Δ⁎. We also show that Editing to a Graph with a Given Degree Sequence has a polynomial kernel when parameterized by k+Δ⁎ if only edge additions are allowed, and there is no polynomial kernel unless NP⊆co-NP/poly for all other combinations of the allowed editing operations
Polynomial Delay Algorithm for Listing Minimal Edge Dominating sets in Graphs
The Transversal problem, i.e, the enumeration of all the minimal transversals
of a hypergraph in output-polynomial time, i.e, in time polynomial in its size
and the cumulated size of all its minimal transversals, is a fifty years old
open problem, and up to now there are few examples of hypergraph classes where
the problem is solved. A minimal dominating set in a graph is a subset of its
vertex set that has a non empty intersection with the closed neighborhood of
every vertex. It is proved in [M. M. Kant\'e, V. Limouzy, A. Mary, L. Nourine,
On the Enumeration of Minimal Dominating Sets and Related Notions, In Revision
2014] that the enumeration of minimal dominating sets in graphs and the
enumeration of minimal transversals in hypergraphs are two equivalent problems.
Hoping this equivalence can help to get new insights in the Transversal
problem, it is natural to look inside graph classes. It is proved independently
and with different techniques in [Golovach et al. - ICALP 2013] and [Kant\'e et
al. - ISAAC 2012] that minimal edge dominating sets in graphs (i.e, minimal
dominating sets in line graphs) can be enumerated in incremental
output-polynomial time. We provide the first polynomial delay and polynomial
space algorithm that lists all the minimal edge dominating sets in graphs,
answering an open problem of [Golovach et al. - ICALP 2013]. Besides the
result, we hope the used techniques that are a mix of a modification of the
well-known Berge's algorithm and a strong use of the structure of line graphs,
are of great interest and could be used to get new output-polynomial time
algorithms.Comment: proofs simplified from previous version, 12 pages, 2 figure
An FPT 2-Approximation for Tree-Cut Decomposition
The tree-cut width of a graph is a graph parameter defined by Wollan [J.
Comb. Theory, Ser. B, 110:47-66, 2015] with the help of tree-cut
decompositions. In certain cases, tree-cut width appears to be more adequate
than treewidth as an invariant that, when bounded, can accelerate the
resolution of intractable problems. While designing algorithms for problems
with bounded tree-cut width, it is important to have a parametrically tractable
way to compute the exact value of this parameter or, at least, some constant
approximation of it. In this paper we give a parameterized 2-approximation
algorithm for the computation of tree-cut width; for an input -vertex graph
and an integer , our algorithm either confirms that the tree-cut width
of is more than or returns a tree-cut decomposition of certifying
that its tree-cut width is at most , in time .
Prior to this work, no constructive parameterized algorithms, even approximated
ones, existed for computing the tree-cut width of a graph. As a consequence of
the Graph Minors series by Robertson and Seymour, only the existence of a
decision algorithm was known.Comment: 17 pages, 3 figure
Exhaustive generation of -critical -free graphs
We describe an algorithm for generating all -critical -free
graphs, based on a method of Ho\`{a}ng et al. Using this algorithm, we prove
that there are only finitely many -critical -free graphs, for
both and . We also show that there are only finitely many
-critical graphs -free graphs. For each case of these cases we
also give the complete lists of critical graphs and vertex-critical graphs.
These results generalize previous work by Hell and Huang, and yield certifying
algorithms for the -colorability problem in the respective classes.
Moreover, we prove that for every , the class of 4-critical planar
-free graphs is finite. We also determine all 27 4-critical planar
-free graphs.
We also prove that every -free graph of girth at least five is
3-colorable, and determine the smallest 4-chromatic -free graph of
girth five. Moreover, we show that every -free graph of girth at least
six and every -free graph of girth at least seven is 3-colorable. This
strengthens results of Golovach et al.Comment: 17 pages, improved girth results. arXiv admin note: text overlap with
arXiv:1504.0697
Solving Problems on Graphs of High Rank-Width
A modulator of a graph G to a specified graph class H is a set of vertices
whose deletion puts G into H. The cardinality of a modulator to various
tractable graph classes has long been used as a structural parameter which can
be exploited to obtain FPT algorithms for a range of hard problems. Here we
investigate what happens when a graph contains a modulator which is large but
"well-structured" (in the sense of having bounded rank-width). Can such
modulators still be exploited to obtain efficient algorithms? And is it even
possible to find such modulators efficiently?
We first show that the parameters derived from such well-structured
modulators are strictly more general than the cardinality of modulators and
rank-width itself. Then, we develop an FPT algorithm for finding such
well-structured modulators to any graph class which can be characterized by a
finite set of forbidden induced subgraphs. We proceed by showing how
well-structured modulators can be used to obtain efficient parameterized
algorithms for Minimum Vertex Cover and Maximum Clique. Finally, we use
well-structured modulators to develop an algorithmic meta-theorem for deciding
problems expressible in Monadic Second Order (MSO) logic, and prove that this
result is tight in the sense that it cannot be generalized to LinEMSO problems.Comment: Accepted at WADS 201
Surjective H-Colouring : new hardness results.
A homomorphism from a graph G to a graph H is a vertex mapping f from the vertex set of G to the vertex set of H such that there is an edge between vertices f(u) and f(v) of H whenever there is an edge between vertices u and v of G. The H-Colouring problem is to decide whether or not a graph G allows a homomorphism to a fixed graph H. We continue a study on a variant of this problem, namely the Surjective HH -Colouring problem, which imposes the homomorphism to be vertex-surjective. We build upon previous results and show that this problem is NP-complete for every connected graph H that has exactly two vertices with a self-loop as long as these two vertices are not adjacent. As a result, we can classify the computational complexity of Surjective HH -Colouring for every graph H on at most four vertices
Steiner trees for hereditary graph classes.
We consider the classical problems (Edge) Steiner Tree and Vertex Steiner Tree after restricting the input to some class of graphs characterized by a small set of forbidden induced subgraphs. We show a dichotomy for the former problem restricted to (H1,H2) -free graphs and a dichotomy for the latter problem restricted to H-free graphs. We find that there exists an infinite family of graphs H such that Vertex Steiner Tree is polynomial-time solvable for H-free graphs, whereas there exist only two graphs H for which this holds for Edge Steiner Tree. We also find that Edge Steiner Tree is polynomial-time solvable for (H1,H2) -free graphs if and only if the treewidth of the class of (H1,H2) -free graphs is bounded (subject to P≠NP ). To obtain the latter result, we determine all pairs (H1,H2) for which the class of (H1,H2) -free graphs has bounded treewidth
Linear MIM-Width of Trees
We provide an algorithm computing the linear maximum induced
matching width of a tree and an optimal layout.Comment: 19 pages, 7 figures, full version of WG19 paper of same nam